Fair student placement

This is a postprint version of the following published document:

Alcalde, J., Romero-Medina, A.,

(20

17

).

Fair student placemen

.

Theory and Decision

,

v.

83

, n.

2

,

pp.

293-307

. Available in:

https://doi.org/

10.1007/s11238-017-9598-8

Fair student placement

José Alcalde1 · Antonio Romero-Medina2

Abstract We revisit the concept of fairness in the Student Placement framework. We declare an allocation asα-equitable if no agent can propose an alternative allocation that nobody else might argue to be inequitable. It turns out thatα-equity is compatible with efficiency. Our analysis fills a gap in the literature by giving normative support to the allocations improving, in terms of efficiency, the Student Optimal Stable allocation.

Keywords Student placement·Fair matching

1 Introduction

This paper explores the trade-off between efficiency and equity in the Student Place-ment problem and provides a new ‘compromise’ solution for this family of models that always select, at least, one allocation.

This paper supersedes our previous version titled “Re-Reforming the Bostonian System: A Novel Approach to the Schooling Problem,” (Alcalde and Romero-Medina 2011). The authors acknowledge César Alonso-Borrego, Salvador Barberà, Estelle Cantillon, Aytek Erdil, Eun Jeong Heo, Fuhito Kojima, Jordi Massó, Thayer Morrill, Josep E. Peris, Pablo Revilla and José María Zarzuelo, two referees and the Associate Editor for their useful comments. We would also like to acknowledge the financial support of the Spanish Ministry of Economy and Competitiveness under projects ECO2013-43119-P and ECO2016-77200-P (Alcalde), MDM 2014-0431 and ECO2014-57442-P (Romero-Medina). Antonio Romero-Medina also acknowledges MECD PR15/00306 and the hospitality of the Department of Economics at Boston College.

B

1 _{IUDESP, University of Alicante, Sant Vicent del Raspeig, Spain} 2 _{Universidad Carlos III de Madrid, Getafe, Spain}

Student Placement mechanisms were modeled in Balinski and Sönmez (1999), inspired by the two-sided problems introduced byGale and Shapley(1962). These authors explore how school seats should be distributed among the students. A spe-cific, and thus differential, characteristic of this problem is that schools are modeled so as to capture social conventions on commonly accepted primitives of equity. These social agreements are captured in the description of how schools prioritize the dif-ferent students.1The connection between Student Placement and two-sided matching problems has inspired some authors to propose allocation systems to improve upon the ones that were established in some geographical areas (Abdulkadiro˘glu and Sönmez 2003) and/or to identify the problems that current Student Placement systems might exhibit, and thus propose how to avoid them.

One of the main problems faced by the Student Placement systems is the existence of an equity–efficiency trade-off (see, e.g., Example 1 inAbdulkadiro˘glu and Sönmez 2003). The persistence of this conflict is reflected in the proposal of two (incentive compatible) procedures to distribute the available seats among the newcomers. The first one, named the Top Trading Cycles mechanism, TTC hereafter, ensures allocative efficiency at the expense of equity. The second procedure, known as the Student Optimal Stable mechanism, seeks to ensure equity at the cost of reducing the efficiency of the allocation. SinceAbdulkadiro˘glu and Sönmez(2003) the literature has proposed a few ways to reduce the relevance of such a conflict. In parallel with this normative approach, it has become commonly known that any attempt to reduce the efficiency– equity collision conflicts with the design of incentive compatible mechanisms (see, e.g.,Kesten 2010, Proposition 1).

In this matter, Kesten (2010) resorts to the idea of ‘consent’ by some students to avoid the potential presence of some inefficiency. The interpretation of Kesten’s consent is very related to the algorithm he describes to reduce the relevance of this trade-off between the two properties above, namely efficiency and equity. Under the Efficiency-Adjusted Deferred Acceptance algorithm, EADAM hereafter, the students may consent to waive her priority over some schools. A waiver by some student holds whenever two conditions are fulfilled. The first one is that, by waiving the priority, she does not hurt herself; the second one is that by waiving, the assignment of other students improves. This idea of consent has been also employed by Tang and Yu

(2014) to introduce an algorithm that is computationally simpler than the EADAM. As Kesten (2010) andTang and Yu (2014) show, their algorithms select efficient allocations that minimize the presence of inequity when all the students consent to waive their priorities. The lack of a rationale behind the consenting process prevents EADAM to substitute the role of a fairness concept overcoming the efficiency–equity trade-off.

Recently,Morrill(2015) has concentrated on procedural equity, rather than alloca-tive equity. His equity notion can be described as follows: imagine that we are employing a specific mechanism. As a consequence, Abel obtains a seat at School 1. Then, Beth argues that it is unfair because she prefers to be allocated at School 1 1 _{The main aspects affecting these priorities are(1)the distance between each student’s residence and} where the school is placed; and(2)the number of siblings attending the school (see, e.g.,Abdulkadiro˘glu and Sönmez 2003).

rather than being at her actual school. Beth’s objection is disregarded whenever there is another student who might obtain the seat Abel was assigned to by misreporting her preferences.Morrill (2015) proves that the TTC is the only mechanism that is strategy-proof, efficient and fulfills his equity notion (seeMorrill 2015, Definitions 1 and 2). InMorrill(2015), the conditions that allow an objection—to an allocation—to be admissible is very related to the procedure used to select the allocation. This induces that some of the placements sanctioned as just are hardly identifiable as equitable (see Example2).

In a framework close to ours, Abdulkadiro˘glu and Sönmez (1998) propose an alternative way to circumvent the efficiency–equity trade-off. They exploit the fact that some students declare preferences for schools that are not in their area of residential priority. They propose the following procedure to combine efficiency and equity: first, to ensure efficiency, they allow students to sequentially select their preferred school among those with vacant positions. Then, since this procedure is very dependent on the ordering in which the students make their choices—and thus very inequitable— select the ordering in which the selections are made at random, by drawing a fair raffle among all the possible orderings. This mechanism, known as the Random Serial Dictatorship, combines ex-post efficiency and ex-ante equity. Nevertheless, as pointed out byBogomolnaia and Moulin(2001), the Random Serial Dictatorship mechanism fails to be ex-ante efficient. Furthermore,Bogomolnaia and Moulin(2001) establish the incompatibility of the two appealing normative properties from an ex-ante perspective. Our approach in the present paper departs from the above-mentioned analysis. Our aim is not to propose a systematic way to select efficient allocations that are not questionable in terms of equity.2Our objective is to find a weakening of equity that turns out to be compatible with efficiency.

To describe how we reach our target it might be relevant to go back to the origins of this literature. As we pointed out, the growth of the literature on Student Placement is related to the analysis of two-sided matching mechanisms. The idea of justified envy, as defined by Balinski and Sönmez (1999), is borrowed from the notion of pairwise stability introduced byGale and Shapley(1962). This connection invites to explore how some classical ideas of stability, weaker than that of the core, might be re-interpreted in terms of weak equity in the Student Placement framework to elude the disconnection between efficiency and (weak) equity. Therefore, we discuss how to state whether an individual’s objection is admissible.3

In this paper, we propose a notion of weak equity that follows the idea of absence of envy introduced byFoley(1967).4 To illustrate whether an allocation is weakly equitable or not, let us assume that this allocation has been proposed. A student can

2 _{For completeness we will describe, in the Appendix, one of such procedures.}

3 _{This approach, introduced byAumann and Maschler(1964), is in the origin of a wide literature studying} different weak notions of stability in co-operative games. In particular, the logic process described inZhou

(1994) to define its Bargaining Set is the closest to our notion of equity. Related to stability notions in marriage problems,Klijn and Massó(2003) propose a weak stability notion also inspired by the Zhou’s Bargaining Set.

4 _{Remark1discusses the main differences between justified envy, as defined byBalinski and Sönmez} (1999), and our weak equity.

object to this allocation by proposing an alternative allocation assigning her to a school where she has priority (with respect to some of the students previously assigned to that school). Then, the society should evaluate whether the objection is accepted or not. The proposal is defeated, and thus the objection is not accepted, whenever some student can present a new objection to the previous one formulated in the same terms. Otherwise, the objection is admissible. We consider that an allocation is almost-equitable, orα-equitable henceforth, whenever no student is able to object to it, by proposing an alternative allocation which cannot be objected in the same terms. We prove, Theorem1, the existence of efficient,α-equitable allocations. Recall that under equity, as defined byBalinski and Sönmez(1999), such an existence cannot be granted. A further question that we deal with is how strong our equity property is. In Theorem2

we show thatα-equity is a weaker condition of equity, which overcomes the trade-off with efficiency: whenever we restrict to efficient allocations,α-equity and equity coincide unless no equitable allocation exists.

The remaining of the paper is organized as follows: Sect.2introduces the basic model.α-equity is defined in Sect.3, which also contains our main results. Finally, Sect.4concludes. The proofs are gathered in the Appendix.

2 The framework

A group ofn students has to be distributed amongmdifferent schools. The setI =

{1, . . . ,i, . . . ,n}stands for the set of students, whereasS= {s1, . . . ,sj, . . . ,sm}

describes the set of schools. We consider the existence of an outside optionso, which

is interpreted by each student as the possibility of not attending any school inS. Each student, sayi, orders the schools according her own preferencesi which

describes a linear preordering in S∪ {so}.i representsi’s weak preferences, i.e.,

sj i shdenotes that eithersj is preferred toshunderi’s preferences orsj =sh.

Each school, say sj ∈ S, has a fixed capacity qj ≥ 1 denoting its number of

available seats, and interpreted as the maximum of students it can enroll. To guide potential admissions procedures, each school determines how to prioritize students through a linear preordering inI.Pj denotes the priority list bysj.5

A Student Placement problem, or simply a problem, can be synthetically described as P ≡ {(I,);(S,P,Q)}, where ≡ (i)i∈I is the preferences profile; P ≡

(Pj)sj∈Sis the vector describing each school priority list; andQ≡(qj)sj∈S summa-rizes the capacities for each school.

A solution to our problem, or matching, is a distribution of the seats among the students. It is formally described through a correspondenceμ: I∪S I∪S∪ {so}

such that

(a) for eachi ∈ I,μ(i)∈S∪ {so};

(b) for eachsj ∈S,μ(sj)⊆I, and|μ(sj)| ≤qj; and

(c) for eachi ∈ Iandsj ∈ S,μ(i)=sj if, and only if,i ∈μ(sj). 5 _{For completeness, we consider that the number of seats that the outside option has isq}

o≥n. Sinceso

The central normative properties, related to a matching, in the Student Placement framework are efficiency and equity, as described below.

Given a problemPand two different allocations for it, sayμandμ, we say thatμ dominatesμ wheneverμ(i)i μ(i)for each studenti whose allocation differs.μ

isefficientif there is no matching dominating it.

In the Student Placement framework, the notion of equity follows the original description byFoley(1967), in the sense that it is required that no student is envied by anyone else. The existence of justifiable envy requires the coincidence of two facts. The first one is the presence of some student expounding her envy. The second one is the consent of the school proposed by this student.

We say that studentienviesi at matchingμwhenever there is a school, saysj ∈S,

such thatsj =μ(i)and

sj i μ(i). (1)

This envy is justifiable whenever Condition (1) is fulfilled and

i Pji . (2)

Matchingμisequitableif no student is justifiably envied by someone else. Finally, we say that matchingμisfairwhenever it is both efficient and equitable. The set of fair allocations for problemPis denoted asF(P).

Remark 1 The notion of equity we introduce below differs from that proposed by

Balinski and Sönmez(1999). These authors also consider two additional sources of inequity of allocationμ:

(i) The absence of individual rationality from some student’s point of view, i.e. the presence of somei such thatsoi μ(i); and

(ii) The existence of an unoccupied seat at some desired school, i.e. the existence of somesj ∈Sandi∈ I such that|μ(sj)|<qj andsj i μ(i).

Note that the two possibilities above are related to the lack of efficiency rather than to the presence of envy between students. Nevertheless, given that we are interested in allocations combining efficiency and equity, our conception of fairness coincides with that ofBalinski and Sönmez(1999).

The existence of efficient or equitable allocations is easily granted. In particular, the TTC algorithm (Abdulkadiro˘glu and Sönmez 2003) always produces efficient allocations and the Deferred Acceptance algorithm (Gale and Shapley 1962) associates an equitable matching, known as the Student Optimal Stable matching and denoted asμSO, to each Student Placement problem.

Unfortunately, there are instances where efficiency and equity become incompati-ble, as pointed out in the next example.

Example 1 Consider the following problem involving three students and three schools.

I = {1,2,3},S = {a,b,c}. Each school has one available seat, i.e.qj =1 for each

123 a c c c a a b b b Pa Pb Pc 3 2 1 2 1 3 1 3 2 This problem has four efficient matchings:

123

μA _{a b c}

μB _{a c b}

μC _{b a c}

μD _{b c a}

Note that none of these matchings is equitable. This is because, (a) at matchingμAstudent 2 justifiably envies 1;

(b) at matchingμBstudent 3 justifiably envies 1; (c) at matchingμCstudent 1 justifiably envies 3;and (d) at matchingμDstudent 1 justifiably envies 2. Therefore, this problem has no fair allocation.

3 Almost-equitable allocations

As we mention in Example1above, the problem we considered has no fair allocation. In particular matchingμAfails to be equitable. The reason is that student 2 justifiably envies 1. We can interpret 2’s objection toμAas the proposal of an alternative alloca-tion, sayμE, fulfilling two properties. First, 2 prefers the school assigned to her under

μE_{rather than the one she obtains underμ}A_{. Second, the aspiration by 2 is supported}

by the school assigned underμE as described below. This objective can be reached by describingμEasμE(1)=b,μE(2)=aandμE(3)=c.

Therefore, we can reconsider how students challenge an allocation to be imple-mented. In such a case an objection of a student to a matching requires the proposal of an alternative allocation. This new proposal must be supported by the school. Then we say that the student objects to the initial allocation through the alternative one. We only consider the objections that are based on the lack of equity. This allows us to redefine equity as the absence ofε-objections, namely objections justified by the lack of equity.

Given a problemP and a matching μwe say that studenti objects in terms of equity toμthroughμ whenever there is a schoolsj ∈ Ssuch that(1) μ(i) =sj,

(2)sj i μ(i)and(3)i Pj i for some studenti ∈μ(sj). In such a case we say that

(i;μ)constitutes anε-objection toμby studenti.

Now, we are reconsidering the arguments above, related to allocationμAin Exam-ple 1. This matching can be taken, in the absence of admissibleε-objections, as a default. The question that we deal with is how to determine whether anε-objection

qualifies as admissible or not. According toZhou(1994) we say that a student’s objec-tion is admissible whenever no one else wishes toε-object to this student’s proposal. For a given problemPand matchingμwe say that anε-objection(i;μ)is admis-sible whenever there is no studenti =iand matchingμ such thati objects in terms of equity toμ throughμ . Otherwise, we say that(i ;μ )constitutes anε -counter-objection to(i;μ).

We can now give a formal definition of what an almost equitable allocation is. Given a problemP, we say that matchingμ isα-equitableif there is no(i, μ)

constituting an admissibleε-objection toμ.

An efficient allocation that is alsoα-equitable is saidα-fair.Fα(P)denotes the set ofα-fair allocations for problemP.

Remark 2 Note that, for a given problem P each equitable allocation is also α -equitable. The opposite is, in general, not true. For instance, matchingμAin Example1

isα-equitable but it fails to be equitable. Note that the unique student that can exert an

ε-objection toμAis 2, who exhibits justifiable envy to 1. Therefore, when objecting to

μ, 2 must propose a matchingμ such thatμ(2)=a. Now, we consider the different possibilities related toμ.

1. μ(3) = c. Then, atμ, 3 justifiably envies 2. Therefore, for anyμ such that

μ (3)=a,(3;μ )constitutes anε-counter-objection to(2;μ).

2. μ(3)=c. Then, atμ, 1 justifiably envies 3. Then, for anyμ withμ (1)=c

we have that1;μ constitutes anε-counter-objection to(2;μ).

Observe that Remark2above also suggest that the efficiency of an allocation might be compatible with itsα-equity. This is asserted in Theorem1below.

Theorem 1 Each problemPhas at least oneα-fair allocation.

The proof of the result above is addressed in the Appendix. It is built in a construc-tive way that exhibits some similarities with Varian’s proof of the existence of a fair allocation in distributive economies (seeVarian 1974). Our constructive proof can be synthesized as follows: first, compute an equitable—and thusα-equitable—allocation. Select the Student Optimal Stable allocation. Once each student is allocated a seat, this can be interpreted as her initial endowment in an exchange economy. Since money does not play an active role in our model, this economy is reinterpreted in terms of a housing market (Shapley and Scarf 1974). By identifying an equilibrium in this market, we obtain an efficient allocation. This is done through the application of the Gale’s Algorithm.6We finally show that this efficient matching is alsoα-equitable.

It is well known that for any given problemPthe set of its fair allocations is either a singleton or the empty set. As Theorem 1 reports, there is always an allocation fulfilling α-equity. A way to measure how much the equity requirement has been relaxed, when adoptingα-equity instead of equity, comes from comparing the sizes of the two sets of fair andα-fair allocations, when the former is non-empty. This comparison is straightforwardly derived from our Theorem2below.

6 _{This algorithm was introduced inShapley and Scarf(1974) under the name of Top Trading Cycle} algo-rithm. SinceAbdulkadiro˘glu and Sönmez(2003) refers a similar, but different algorithm by using the same name, we prefer to call it Gale’s algorithm to avoid confusion.

To be precise, Theorem2allows describing the set ofα-fair allocations that each problem exhibits as the efficient matchings dominating the Student Optimal Stable matching, except when such an allocation is efficient itself.

Theorem 2 LetPbe a problem. Matchingμis anα-fair allocation forPif,and only if,μis efficient and for each student,say i,μ(i)i μSO(i).

The proof of the above result is relegated to the Appendix.

We conclude this section by introducing two direct implications from Theorem2

above. The first one, Corollary1, establishes that the Student Optimal Stable match-ing is the unique allocation (if any) combinmatch-ing efficiency and equity. The second one, Corollary2, reports that when a fair matching exists, the sets of fair andα-fair allo-cations coincide. Notice that this implies, in particular, that even thoughα-fairness is weaker than fairness, it coincides with standard fairness on a restricted domain (i.e. the domain where the standard equity and efficiency properties are compatible).

Corollary 1 For each problemP,F(P)⊆ {μSO(P)}.7

Corollary 2 For each problemPsuch thatμSO(P)is efficient,F(P)=Fα(P).

4 Concluding remarks

In this paper, we explored the compatibility of weak notions of equity with allocative efficiency in the Student Placement problem. Our first concern is to adapt the classical proposal byFoley(1967) that identifies equity of an allocation with the absence of envy, i.e. no student envies someone else’s seat.

Our approach distinguishes between objections in terms of Pareto improvements and those that are justified because of merely inequality aspects. This distinction sim-plifies a proper normative analysis based on efficiency and/or equity from an allocative perspective.

Since efficiency and equity are two appealing properties, but incompatible in our framework, we investigate how the equity notion might be relaxed to circumvent such incompatibility. By adapting the idea of ‘resentment’ (seeRawls 1971, pg. 533), we propose a notion of ‘almost’-equity that can be justified as follows: when a student claims that the allocation fails to be equitable, she must propose an alternative allo-cation. The new proposal must match two natural properties. The first one is that the claimant should benefit when adopting her proposal. Otherwise, what she is proposing, if accepted, harms her own interests. Clearly, it is hard to expect a student to object an allocation, when such a demand opposes her well-being. The second restriction is that the new proposal overcomes the students’ possible claims in terms of equity. Note that, otherwise the proposing student can be criticized because what she suggests is to proceed unfairly. Note that, when a student objects because of equity reasons, the Golden Rule (or ethic reciprocity) ‘do not do unto others what you would not want done to yourself’ applies.

7 _{We employ here, as well as in Corollary2, the expressionμ}SO_{(P)rather than the usualμ}SO_{just to} highlight that this matching is referred to the problemP.

We find that our notion ofα-equity is compatible with the efficiency requirement. More than that, we demonstrate that when efficiency and equity are compatible,α -equity and -equity are equivalent across the set of efficient allocations. As a consequence of that, our results complement the analysis of other authors worried about the equity– efficiency pairing,

(a) for any problemP, the outcome of the EADAM (Kesten 2010) isα-fair when all the students consent;

(b) for any problem satisfying the acyclicity condition (Ergin 2002), there is a unique

α-fair matching. It is its Student Optimal Stable matching.

As we mentioned in the Introduction, this is the first paper to suggest a weakening of equity that is compatible with efficiency on the whole domain of student placement problems.Morrill(2015) analyses a kind of procedural equity. His definition of a just allocation does not embody our idea of fairness. As can be seen in Example2, the TTC algorithm fails to select either fair orα-fair allocations.

Example 2 Consider problem P involving three students, I = {1,2,3} and three schools,S = {a,b,c}, having a vacant seat each. The next table summarizes students’ preferences and school priorities.

123 c a a b b b a c c Pa Pb Pc 1 2 3 2 1 2 3 3 1

When applying the TTC to this problem we obtain matchingμTTC, withμTTC(1)=c;

μTTC_{(2)=b; andμ}TTC_(3)=a.

Note that any efficient matching must allocate 1 atc. This is becausecis the best school for 1 and the worst for the remaining students. Therefore, the conflict in which students 2 and 3 incur—because both of them want to be allocated at schoola- is elucidated in accordance with Pa. As a result, a is assigned to 2. Notice that this

rationing yields to describe the unique fair—and thusα-fair—matching,μSO, such thatμSO(1)=c;μSO(2)=a; andμSO(3)=b. It can be also verified that(2;μSO) constitutes an admissibleε-objection toμTTC, pointing out the lack of equity exhibited by the latter matching.

To conclude, we want to suggest an open question related to the design of mech-anisms implementingα-fair allocations. There are some authors pointing out that no strategy-proof mechanism dominates the Student Optimal Stable mechanism (see, e.g.

Abdulkadiro˘glu et al. 2009;Kesten 2010;Kesten and Kurino 2016). Therefore, our Theorem2allows determining that no incentive compatible mechanism selectsα-fair allocations. In this line,Alcalde and Romero-Medina(2015) explore the existence of restrictions on students’ preferences where such mechanisms exist. Their findings are not much encouraging. If no condition is imposed on school priorities, manipulability of mechanism selectingα-fair allocations is guaranteed unless the students’ admissi-ble preferences satisfy the (restrictive)β-Condition. Hence, as a natural complement

to our results, it might be interesting to ask about the existence of mechanisms whose expected outcome—given that students should behave strategically—isα-fair.

Appendix

Appendix A: Existence ofα-fair allocations

As we anticipated in Sect.3we proceed to prove Theorem1in a constructive way. The process can be described as follows: consider a given Student Placement problem

P, and a matchingμ. By interpretingμas an initial endowment for each student, we can understand that(P;μ)constitutes a ‘seating market’ where students are allowed to exchange the seats they have been allocated. This market exhibits some similarities with the ‘housing market’ introduced byShapley and Scarf(1974). Therefore, the tools usually employed to solve housing problems might be useful to explore how exchanges are conducted in the seating markets.

Consider a fixed problemP and compute its Student Optimal Stable matching,

μSO_{. It can be obtained by applying the Deferred Acceptance algorithm (Gale and} Shapley 1962). Associated with the pair(P, μSO_{)we describe, for each student, her}

preferences for exchanging, denotedEi as the linear ordering onIdefined as follows:

(a) For each two studentshandksuch thatμSO(h)=μSO(k),h Ei kif, and only

if,μSO(h)i μSO(k); and

(b) For each two distinct students h and k such thatμSO(h) = μSO(k) = sj ∈

S∪ {so},h Ei kif, and only if,h Pj k.8

Given the preferences for exchanging associated with each student, E =(Ei)i∈I

describes a profile of preferences for exchanging.

Note that the pair(I;E)accommodates the structure of a housing market (Shapley and Scarf 1974). We now describe how to Gale’s algorithm is applied to this problem so to calculate the unique core allocation for this market:

Step 1. Build a directed graph whose nodes are the agents inI. This graph hasnarcs connecting each student with her preferred ‘mate for exchanging,’ i.e., for eachi ∈ I, there is an arc fromi, pointing the maximal onI accordingEi.

Since there is exactly one arc starting at each of thennodes, this graph must have at least one cycle.9Moreover, no student is involved in two different cycles. Then, each student belonging to a cycle is definitively assigned the seat she is pointing in the cycle,10_{and leaves the market. Letμ}ε_{(i)denote the}

school assigned toi, when she is in a cycle.

LetI1be the set of students not belonging to a cycle. Then, if I1is empty, 8 _{The outcome that we will obtain does not depend on how the outside schools}

oprioritizes the different

students. Nevertheless, and for the sake of completeness, we consider that schoolsoprioritizes the students

according their labels; i.e.i Pohwheneveri<h.

9 _{A cycle is a set of students,i}

1, . . . ,ik, . . . ,ir, such that for eachk, 1≤k<r, there is an arc connecting

iktoik+1; andiris connected toi1. Note that a cycle might involve a unique student.

10 _{Note that we identify each student with the seat she obtains atμ}SO_{. Therefore, wheniis pointinghwe} interpret thatiwants to obtain a seat at schoolμSO(h).

the algorithm terminates producing matchingμε, previously described. Oth-erwise, go to step 2.

. . .

Step t. A graph involving the students inIt−1is generated. The nodes coincide with these students. There is an arc connecting each student inIt−1to her preferred mate for exchanging, among the ones that are still in the market, according her preferences for exchangingEi. As in the previous step, this graph has at least

one cycle. Each student involved in a cycle is assigned the seat she is pointing to and leaves the market. LetIt be the set of students inIt−1being not in a cycle in the graph built in this step. IfIt is empty, the algorithm terminates producing matchingμε, described throughout steps 1 tot. Otherwise, go to stept+1.

Note that, since for eacht such that It _{= ∅,I}t _It−1_{, the algorithm ends in finite}

steps.

Previous to demonstrate that matchingμεisα-fair, we will illustrate how to compute it through an example:

Example 3 Consider problemPinvolving 8 students, I = {1, . . . ,i, . . . ,8}and 4 schools,S= {a,b,c,d}, having 2 vacant seats each. The students’ preferences are

12345678

b c c d a a a b a a b b c b b a c d a c d c d c d b d a b d c d

The priorities of the schools are

Pa Pb Pc Pd 4 3 7 5 1 2 5 6 2 7 6 3 6 6 8 8 8 4 2 2 7 1 3 4 5 8 1 7 3 5 4 1

We first compute the Student Optimal Stable matching,μSO. It is calculated by apply-ing the Deferred Acceptance algorithm. At each step, each student applies for her preferred school—among the ones not having rejected her previously—and each school rejects the less prioritized students among those sending it an application,

so to keep all its seats filled. The next table describes, for each step, the applications that each school receives, and which are accepted by the school.

Step a b c d so μt(a) μt(b) μt(c) μt(d) μt(so) 1 5,6,7 1,8 2,3 4 − 6,7 1,8 2,3 4 5 2 6,7 1,8 2,3,5 4 − 6,7 1,8 2,3 4 3 3 6,7 1,3,8 2,5 4 − 6,7 1,3 2,5 4 8 4 6,7,8 1,3 2,5 4 − 6,8 1,3 2,5 4 7 5 6,8 1,3,7 2,5 4 − 6,8 3,7 2,5 4 1 6 1,6,8 3,7 2,5 4 − 1,6 3,7 2,5 4 8 7 1,6 3,7 2,5,8 4 − 1,6 3,7 5,8 4 2 8 1,2,6 3,7 5,8 4 − 1,2 3,7 5,8 4 6 9 1,2 3,6,7 5,8 4 − 1,2 3,7 5,8 4 6 10 1,2 3,7 5,6,8 4 − 1,2 3,7 5,6 4 8 11 1,2 3,7 5,6 4,8− 1,2 3,7 5,6 4,8 −

Therefore,μSOis such that μSO(a) = {1,2}, μSO(b) = {3,7},μSO(c) = {5,6} andμSO(d) = {4,8}. Now, we describe the students’ preferences for exchanging. Recall that each student orders I, according the seat each student is allowed under

μSO_{—for instance, since a}

5 b, μSO(1) = a and μSO(3) = b, we have that

1 E5 3—and ties are broken in accordance with the school priorities, for instance,

sinceμSO(1)=μSO(2)=a, and 1Pa2, each studenti should prefer to exchange

her seat to 1 rather than to 2. Summarizing, the preferences for exchange are gathered in the following table:

E1 E2 E3 E4 E5 E6 E7 E8 3 5 5 8 1 1 1 3 7 6 6 4 2 2 2 7 1 1 3 3 5 3 3 1 2 2 7 7 6 7 7 1 5 8 1 5 8 5 8 5 6 4 2 6 4 6 4 6 8 3 8 1 3 8 5 8 4 7 4 2 7 4 6 4

Now, we can run Gale’s algorithm. At the first step, each student points her preferred student for exchanging. As Fig.1shows, this graph has one cycle involving students 1, 3 and 5. Therefore, each of these students obtains a seat at the school that the student she pointed got atμSO. In other words,με(1)=μSO(3)=b;με(3)=μSO(5)=c; andμε(5)=μSO(1)=a.

Once the first step is concluded, and some students leave the market, the second step is similar to the previous one, taking into account that no (remaining) student can select any student that left the allocative process. Now, as showed in Fig. 2, students 2 and 6 are involved in a cycle. This implies thatμε(2)=μSO(6)=c; and

Fig. 1 Gale’s algorithm, first step

Fig. 2 Gale’s algorithm, second step

Fig. 3 Gale’s algorithm, third step

Figure3 captures the graph constructed at the third step. Now, students 7 and 8 exchange the seats they have been allocated atμSO.

Once step 3 is concluded, the unique remaining student is 4. The application of Gale’s algorithm indicates that this student must keep the seat thatμSOassigned to her.

Note that, as illustrated in Example 3, when running Gale’s algorithm, all the students involved in a cycle at the first step obtain a seat at their preferred school. Similarly, all the students involved in a cycle at the second step are allocated a seat at their preferred school, provided that some seats are still unavailable because they have been already allocated at the previous step, and so on. This implies thatμεis an efficient matching.

We can now formally prove Theorem1.

Proof LetPbe a problem andμSOits Student Optimal Stable matching. Note that, when describing each student’s preferences for exchanging we have that for any two studentsiandh,h Ei iwhenever

1. μSO(h)i μSO(i); or

2. μSO(h)=μSO(i)=sj ∈ S∪ {so}andh Pj i.

In particular, this implies that ifμSO(h)=μSO(i)=sj ∈S∪{so}andh Pji, for each

student, sayk,h Ek i. Therefore, two agents having a seat at the same school under

μSO_{will not obtain a definitive seat, when running Gale’s algorithm, at the same step.}

Moreover, since no student leaves the market at some step if she does not belong to a cycle, and each student points her best ‘student for exchanging’ according her

preferences, we have that, for eachi,

με(i)i μSO(i).

Now, assume that με is not α-fair. Since, as previously reported, it is efficient, it must fail to be α-equitable. Then, there should be a student i and a matchingμ such that(i;μ)constitutes an admissibleε-objection toμε. This implies that there is

sj =μ(i)i με(i)i μSO(i), and thussj ∈ S. Therefore, by the Blocking Lemma

(Martínez et al. 2010, Theorem 1), matchingμfails to be equitable, which contradicts thatε-objection(i;μ)toμεwas admissible.

Appendix B: Identifying the set ofα-fair allocations

We now deal with proving Theorem2. It establishes that, for a given problemP, matchingμisα-fair if, and only if, it is efficient and, for each studenti,

μ(i)i μSO(i). (3)

Note that, sinceα-equity implies efficiency by definition, we only need to concentrate on the fulfillment of Condition (3) above.

Proof For a given problemP, letμSObe its Student Optimal Stable matching. Letμ be anα-fair matching. Therefore, it is efficient. Assumeμdoes not fulfill Condition (3) above. Then, there should be a studenti such that

μSO₍

i)i μ(i). (4)

Note that this implies that there is somesj ∈ Ssuch that sj =μSO(i). Otherwise,

μis dominated by matchingμ such thatμ(i) = soand, for each student h = i,

μ(h)=μ(h).

Efficiency also implies that|μ(sj)| =qj. Otherwise,μis dominated by matching

μ _{such thatμ (i) = s}

j and, for each student h = i, μ (h) = μ(h). Moreover,

α-equity ofμimplies that for eachh∈μ(sj),h Pj i. Note that, otherwise,(i;μSO)

constitutes an admissibleε-objection toμ.

Since i ∈ μSO(sj)\μ(sj) and |μ(sj)| = qj, there should be a student h ∈

μ(sj)\μSO(sj). Sinceh PjiandμSOis equitable, it must be the case thatμSO(h)h

μ(h), i.e. Condition (4) is also fulfilled for agenth.

By applying an iterative reasoning, and taking into account that the number of schools is finite, there is an ordered set of students{it}T_t=1and schools{st}T_t=1, with

T ≤m, such that for eacht (moduloT), (a) μ(it)=st;

(b) μSO(it)=st+1; and

(c) st+1it s

t_.

Note that the above implies that μfails to be efficient. In fact, it is dominated by matchingμ such that for eachi ∈ {it}T_t=1,μ(i)=μSO(i)and, for eachh ∈ {/ it}T_t=1,

Now, considerμ, an efficient matching that satisfies Condition (3). Assume thatμ fails to beα-fair. Then, there should be a studenti and matchingμ constituting an admissibleε-objection toμ. This implies that

μ(i)i μ(i)i μSO(i). (5)

Note that, in particular, Condition (5) implies that there is some sj ∈ S such that

sj =μ(i). Therefore, by the Blocking Lemma (Martínez et al. 2010),μ fails to be

equitable. Therefore,(i;μ)fails to be admissible as anε-objection toμ. A

contradic-tion.

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